Knot theory is the study of 3-dimensional shapes obtained by removing loops from the 3-dimensional space. Microscopic examples of knotted loops appear abundantly in solid state physics, biology (notably, the DNA molecule) and various models of quantum computing. Macroscopic knotted loops occur in the shapes of galaxies, black holes and plasma physics. The proposed research focuses on the theoretical aspects of knot theory and more precisely on the relation between classical and quantum knot invariants. Quantum topology offers a classification of knots by various measures of complexity, for example by using the Jones polynomial of a knot (and its parallels). Understanding this filtering leads to deep connections between knot theory and hyperbolic geometry, number theory, complex analysis, algebraic geometry and mathematical physics. An important aspect of this research is the training of graduate students and the collaboration with senior researchers. The research involves difficult numerical experiments and leads to unexpected discoveries of conjectures and proofs. Aside from mathematics and mathematical physics, quantum knot theory is a powerful and effective tool with the potential to impact topological quantum computation and the study of the shape of long molecules.

Classical knot invariants include the incompressible and normal surface invariants by W. Haken, the character varieties of algebraic geometry and mathematical physics and the hyperbolic geometry approach to 3-dimensional manifolds initiated by W. Thurston. The birth of quantum topology was the Jones polynomial in the mid eighties. The discovery of the Jones polynomial attracted mathematicians and physicists of the highest caliber, including the Fields Medalists V. Jones, E. Witten, V. Drinfeld and M. Kontsevich. There are precise, numerically testable conjectures that relate classical and quantum invariants. Progress on these conjectures (which include the Volume Conjecture of Kashaev, the Slope Conjecture and the AJ Conjecture) is the focus of the proposed research. A combination of numerical experimentation with proofs leads to deep connections between low dimensional topology and hyperbolic geometry, number theory, complex analysis, algebraic geometry and mathematical physics. This combination helps to train young researchers, and to make connections with senior researchers in diverse fields.

Agency
National Science Foundation (NSF)
Institute
Division of Mathematical Sciences (DMS)
Type
Standard Grant (Standard)
Application #
1406419
Program Officer
Christopher Stark
Project Start
Project End
Budget Start
2014-07-01
Budget End
2018-06-30
Support Year
Fiscal Year
2014
Total Cost
$377,059
Indirect Cost
Name
Georgia Tech Research Corporation
Department
Type
DUNS #
City
Atlanta
State
GA
Country
United States
Zip Code
30332